Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#392 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2834 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,9)(10,12)(14,19)(15,16)(21,22,24,26)(23,25,28,27);(2,4)(13,14)(15,20)(16,17)(18,19);(1,2)(3,4)(6,9)(10,11)(13,15,18,16)(14,17,19,20)(21,22)(23,27)(24,26)(25,28);(5,7,8,12)(6,10,9,11)(13,16,18,15)(14,17,19,20);(5,6)(7,10)(8,9)(11,12)(13,17)(14,15)(16,19)(18,20)(21,23)(22,25)(24,28)(26,27)Polycyclic presentation:
2^9|a^2=gi,c^2=f,d^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^c=eg110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4214648);